Norm Matloff, July 16, 2022

I had quite a shock to learn that my thesis adviser from my old grad school days, the renowned probabilist Prof. Thomas Liggett, had passed away. He had died relatively young, a brilliant and very decent man, and the news sparked within me some introspection regarding my time working under his direction. The purpose of this note is first to pay belated tribute to Tom, and then to share my thoughts on PhD education, based on my work with Tom so long ago.

Tom's was a giant in the field of probability, and was appointed to the National Academy of Science, one of the very highest honors in science. There were several major obituaries written after Tom's passing, notably here and here. For mathematical content, see especially this writeup. Here I will not duplicate those excellent accounts but will instead write some more personal material, and as noted, use it as a basis for offering some tips to future PhD students and their advisers.

My fellow student Diane Schwartz and I were Tom's first PhD students. He was an Assistant Professor at the time, and thus only a few years older than we were.

In his obituary of Tom, Prof. Rick Durrett, also a renowned probabilist, poses a question: "The psychology of how students choose their advisers is mysterious." It wasn't mysterious at all in my case. Even though my ultimate career goal was statistics rather than probability, I was in awe of Tom's brilliance. I had taken two year-long graduate courses from him, one on real analysis and the second on measure-theoretic probability. To me, there was no question who my first choice for adviser would be.

Although the real analysis course was fairly straightforward, taught out
of Rudin's *Real and Complex Analysis*, it was easily my favorite
among the three courses I took that first year at UCLA. I had attended
my local state college, Cal Poly Pomona, 10 minutes from home, as an
undergraduate. It was a nonresearch institution, and I was struck by
Tom's personal behavior as a mathematician--his bearing, tone of voice
and so on. I was even fascinated by Tom's use of "Mathematics-ese," a
"language" spoken by mathematicians, who use words and phrases such as
*result*, *due to*, and *argument* in ways different from
normal English, e.g. "This result, due to Jones, uses a compactness
argument."

That probability course was an absolute gem. While Tom did nominally follow the standard text by Chung, he had his own alternate proofs for everything, usually using the tools of functional analysis, and always elegant and insightful. At Cal Poly every course was taught straight out of the textbook, so I was absolutely blown away by that probability course. I still am. Tom truly epitomized the notion that good research and good teaching should go hand-in-hand.

This has definitely affected my own teaching and writing. I've been
fortunate to win a couple of teaching awards at UCD, and in the awards
ceremony for the campuswide one, I made a point of citing Tom in my
speech as having deeply influenced my teaching. I've also been fortunate
that my book,
*Statistical Regression and Classification: from Linear Models to
Machine Learning*, was the recipient of the 2017 Ziegal Award.
Though the book is not deeply mathematical at all, I could not resist giving my
own alternate proof of the Tower Property, using Hilbert space
geometry instead of measure theory--very Liggett-esque! I believe many
things in that book can be traced back at least in part to Tom's
influence.

In preparing this essay, I came across wonderful news (though again, belated) of the Liggett Teaching Awards, established initially by a very generous donation by Tom's wife Christina. She had been the department administrative assistant for graduate matters back when I started grad school, the first person I talked to when I arrived. (She married Tom some years later.) So I felt like "the circle had been completed."

I was quite naive about research when I began work with Tom, naivete that was exposed even on my first week at UCLA. The head of the math grad program held some informal meetings with new grad students, three or four at a time. He mentioned that, in his role as head of the program, occasionally a grad student would come to him in great distress. He would try to resolve the matter, an emotionally draining experience for him. Responding to that remark, one of my fellow new grad students asked, "How does that affect your work?" That of course was a reference to research work, but I didn't get it. I thought, "But dealing with student problems IS your work." The ensuing conversation then finally made it clear to me that they were all talking about research.

This then was the my first realization that I had now entered a completely different world than the nonresearch environment I'd had at the state college.

I then entered coursework, treating it as "more of the same" following undergrad work, and to a large extent, it was so. I'd had several professors as an undergrad who assigned very challenging proofs for homework, and this turned out to be excellent preparation for my grad courses at UCLA. I loved the work.

And though some grad students will be horrified to read this, I loved preparing for the PhD qualifying exams. Lots of challenging proofs, of course, which I prepared for by working as many problems as I could get my hands on--in books, old UCLA exams and even old exams at other schools that I wrote away for. The summers of prepping for the exams (four written tests, taken two at a time) were the most enjoyable of my intellectual life to date.

After I approached Tom to express an interest in working with him, he had the traditional math response: He gave me a small research project as kind of a trial run, in which the potential faculty adviser thinks, "Let's see what this student can do." I solved the problem and proudly showed my proof to Tom--only to find he was not completely happy with it. My proof was correct, he agreed, but he had had another path in mind that he thought was better. Viewing the incident today as a computer science professor , it is a bit ironic, as my solution was recursive, a technique common in computer science. But at any rate, this first introduction to doing research under Tom's high standards was an eye opener.

Tom was interested in a research area known as *infinite particle
systems*, an extremely abstract and technically difficult stochastic
process model inspired by (though much more general than) statistical
physics. Even proving the existence of the process involves some very
advanced machinery! (Developed by Tom and UCLA colleague
Michael Crandall
.)

The field itself had originated from the legendary Frank Spitzer. Among other things, Spitzer had devised a clever way to analyze infinite particle systems, via an auxiliary Markov chain. This transformed an intractable uncountably infinite problem to a tractable countably infinite one. The reader of this note need not understand those mathematical terms, but just keep the point in mind that Spitzer's device is the key to analyzing infinite particle systems. And it will be key in the thoughts I share below on PhD education.

At the time, Tom had just published his first major work in the infinite particle systems area, on what he called the "voter model." Instead of thinking of the particles in physics terms, say each one having a spin in one of two directions, Tom used a "voter" metaphor. Each "voter" changes his/her political stance at random times, but with some substantial likelihood of following the example of the voter's neighbors.

Ever the rebel, I proposed to model a setting in which each voter tends
to vote the *opposite* of his/her neighbors. Tom liked the idea
(he even mentioned that his nonmathematician wife liked the idea), so
I started pursuing it.

However, I was totally ignorant of an extremely basic principle of research: One builds on the previous work of others. Somehow I'd never realized this basic tenet. I thus wrongly approached research as just another course homework problem, to be solved largely from first principles rather than on leveraging previous literature. I had started reading Tom's papers, finding them tough going, especially after I encountered an error (which Tom confirmed). So I then did a cursory reading, if that, and none at all of Frank Spitzer's work.

What I urgently needed was to use Frank's device of the auxiliary Markov
chain. Without it, I spent several unproductive months on wild goose
chases, trying to develop my own *ad hoc* tools. Meanwhile, Tom
had a Sloan Fellowship and was constantly traveling. No e-mail in those
days, so he and I communicated only occasisonally by "snail mail."
Eventually he asked, fortunately very tactfully, "Why aren't you using
the auxiliary Markov chain?", and I then finished most of the
dissertation in a few months. Duh.

I published two papers on my dissertation work, viewable here and here . I then moved on to statistics, e.g. this and eventually computer science, e.g. this.

*Suggestions for PhD students and their advisers:*

In advising students who do research with me, whether they be at the PhD, Master's or even undergraduate level, I assume they have talent and interest, but otherwise I make no assumptions, always aware that some students may be as clueless as I was.

In most cases in which an undergrad does research with me, they are thinking of grad school. I highlight this "different world" point, which most are not aware of. Unlike the undergrad level, where many students choose a major for largely practical reasons, most students in a good grad program have a keen intellectual curiosity and interest in the subject matter. That already makes it a different world! Similarly, I explain that in contrast to undergrad work, in which the professor provides lots of structure, in PhD research the student to a large extent formulates his/her own problems.

Given my own dissertation history of ignoring the research literature, I remind them of the phrase, "Standing on the shoulders of giants," emphasizing the importance of a thorough literature search. Fortunately, Google makes that easier these days. I also warn them that many research papers are poorly written, and give them tips for picking out the major points. I stress that although the more independence the student has in his/her work, the better, it is essential to keep constant contact with the adviser.

In addition, I talk about the practical aspects of publishing, e.g. how to make one's writing enticing, how to judge the quality of a journal or conference and so on. Actually, as I have worked in several fields, I feel I myself am still learning these things! I also am very frank with them in explaining that the quality of research reviewers varies greatly, and that one must learn to deal with rejection.

Though quite caring, Tom was a rather reserved man. In addition, his brilliance could be a bit intimidating. He and I were never close, and I believe that may have been true for some of his other students, at least for us early ones. We had worked with him for quite some time before he shared his amazing background---he had grown up in South America in a missionary family, was fluent in Spanish and so on. Who knew? I'm told that he did become closer with later students.

I do try to connect with my students on a personal level. I consider my research students to be my colleagues and friends, in spite of our having different academic statuses, being in different life stages and so on. I try to put myself into their shoes, and hope they have some understanding of being in mine. Finally, I value that I learn as much from them as they learn from me.

Tom and I lost touch over the years, but in 2009 I had a delightfully
surprising encounter with a recent paper of his. At that time, a number
of us at UCD, from several quite varied departments, had formed an
informal group on the subject of random networks. This is the "Six
degrees of separation" field, often used to model social networks. One
day we held a video conference with researchers at several UC campuses,
and someone from the UCLA Sociology Dept. was speaking. Suddenly, out
of the blue, the speaker mentioned a theorem Tom had proven in support
of the speaker's sociological model! This was a pleasant surprise, but
actually to be expected as the natural evolution of Tom's voter models.
It turned out that Tom had been working in the field since at least far
back as 2004 (An Infinite Stochastic model of Social Network Formation,
*Stoch. Procs. and Their Apps.*, with S. Rolles).

I guess that lack of contact with Tom is reflected in the fact that I did not learn of his passing until two full years later. I deeply regret not ever telling him of his profound influence on me, which continues to this day.

Tom once mentioned a researcher who had claimed some impressive research result that had been questioned by, among others, Paul Lévy, one of the pioneers of modern probability theory. Tom told me, "Well, Lévy was someone to be reckoned with, so this researcher hastily rechecked his result. etc." That phrasing stuck with me, and I must say that Tom was definitely someone to be reckoned with.